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REDUCTS IN INCOMPLETE INFORMATION SYSTEMS

REDUCTS IN INCOMPLETE INFORMATION SYSTEMS. Zbigniew Ras. Information Systems. S = (X , AT ) is an information system, where X - objects, AT -attributes ( partial functions from X into 2 Va  {*} ) , V a - set of values of attribute a. Example 1:

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REDUCTS IN INCOMPLETE INFORMATION SYSTEMS

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  1. REDUCTS IN INCOMPLETE INFORMATION SYSTEMS ZbigniewRas

  2. Information Systems S = (X, AT) isan information system, where • X - objects, • AT-attributes (partial functions from X into 2Va {*}), • Va - set of values of attribute a.

  3. Example 1: S = ({1,2,3,4,5,6}, {Price, Mileage, Size, Accident}) defined below: • Let A  AT. By similarity relation based on A we mean: •  SIM(A) = {(x,y)  XX: (a A)[a(x)  a(y)  or a(x) = * or a(y) = *]}. •  SIM(A) is a tolerance relation (reflexive, symmetric). •  Let IA(x) = {yX: (x,y)  SIM(A)} - tolerance class for x with regard to A. •  X/SIM(A) = {IA(x) : x X} – not a partition of X in general.

  4. Definition: A  AT is a reductof information system S = (X, AT) iff • SIM(A) = SIM(AT) and • (BA)[SIM(B)  SIM(A)].  A  AT is a reduct of information system S=(X, AT) for x iff • IA(x) = IAT(x) and • (BA)[IB(x)  IA(x)].  In our example {Price, Size, Accident} is a reduct of S.

  5. Decision Systems • S = (X, AT {d}) decision system, where X- objects, AT - classification attributes, d - decision attribute, where d(x) Vd (value is certain). • Let A  AT and A(x) = {v : d(y) = v and y  IA(x)} /generalized decision in S/

  6. Example 2:Decision System S with “generalized decision” as the extra feature. Definition: Set A  AT is a reduct of S (relative reduct or d-reduct), iff A = AT and (BA)[ B A ]. Set A  AT is a reduct of S for x  X (relative reduct for x or d-reduct for x) iff A(x) = AT(x) and (BA)[ B(x)A(x) ].

  7. Example: • {Size, Accident} is a is a relative reduct of S . • {Size, Accident} is a relative reduct of S for object 3 and • {Price, Accident} is a relative reduct of S for object 2. Relative reducts for objects are used to construct rules: • (Price, low)  (Accident, engine)  (d, good) • (Size, compact)  (Accident, doors)  (d, poor) • (Size, full)  (d, good)  (d, excel)

  8. Computation of Reducts: Discernibility table Here we use P – Price, M – Mileage, S – Size, A- Accident. Discernibility Function: F(P,M,S,A) = PSA(PA) = PSA – (reduct) Reducts for objects: F(1)=PS, F(2)=PA, F(3)=SA, F(4)=PS, F(5)=SA, F(6)=PS.

  9. Discernibility table Computing d-Reducts: Here we use P – Price, M – Mileage, S – Size, A- Accident. Discernibility Function: F(P,M,S,A) = PSA(PA) = PSA – (d-reduct) d-reducts for objects: F(1)=PS, F(2)=PA, F(3)=SA, F(4)=(PA)S, F(5)=SA, F(6)=S.

  10. Thank You !

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